On the local convexity of intersection bodies of revolution
نویسندگان
چکیده
One of the fundamental results regarding intersection bodies is Busemann’s theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is natural to ask how much of convexity is guaranteed under the intersection body operation. In this paper we provide several results about the (strict) improvement of convexity for intersection bodies of symmetric bodies of revolution. We show that the intersection body of a symmetric convex body of revolution always has a ‘very round’ boundary near the equator. Here, ‘very round’ can be expressed by the modulus of convexity around the equator. This result can be applied to get the statement that the double intersection body of a convex body of revolution is close, in the Banach-Mazur distance, to an ellipsoid in high dimension. We also prove results on the local convexity around the equator of the intersection body of a star body of revolution.
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